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| ===Commentary Note 11.7=== | | ===Commentary Note 11.7=== |
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− | <pre>
| + | We come now to the very special cases of 2-adic relations that are known as functions. It will serve a dual purpose on behalf of the exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the RIG material. |
− | We come now to the very special cases of 2-adic relations that are | |
− | known as functions. It will serve a dual purpose on behalf of the | |
− | exposition if we take the class of functions as a source of object | |
− | examples to clarify the more abstruse concepts in the RIG material. | |
| | | |
| To begin, let's recall the definition of a local flag: | | To begin, let's recall the definition of a local flag: |
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| + | <pre> |
| L_x@j = {<x_1, ..., x_j, ..., x_k> in L : x_j = x}. | | L_x@j = {<x_1, ..., x_j, ..., x_k> in L : x_j = x}. |
| + | </pre> |
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− | In the case of a 2-adic relation L c X_1 x X_2 = X x Y, | + | In the case of a 2-adic relation L c X_1 x X_2 = X x Y, we can reap the benefits of a radical simplification in the definitions of the local flags. Also in this case, we tend to denote L_u@1 by "L_u@X" and L_v@2 by "L_v@Y". |
− | we can reap the benefits of a radical simplification in | |
− | the definitions of the local flags. Also in this case, | |
− | we tend to denote L_u@1 by "L_u@X" and L_v@2 by "L_v@Y". | |
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− | In the light of these considerations, the local flags of | + | In the light of these considerations, the local flags of a 2-adic relation L c X x Y may be formulated as follows: |
− | a 2-adic relation L c X x Y may be formulated as follows: | |
| | | |
| + | <pre> |
| L_u@X = {<x, y> in L : x = u} | | L_u@X = {<x, y> in L : x = u} |
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| = the set of all ordered pairs in L incident with v in Y. | | = the set of all ordered pairs in L incident with v in Y. |
| + | </pre> |
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− | A sufficient illustration is supplied by the earlier example E. | + | A sufficient illustration is supplied by the earlier example ''E''. |
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| + | <pre> |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| o o o o o o o o o o X | | o o o o o o o o o o X |
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| o o o o o o o o o o Y | | o o o o o o o o o o Y |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| + | </pre> |
| | | |
| The local flag E_3@X is displayed here: | | The local flag E_3@X is displayed here: |
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| + | <pre> |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| o o o o o o o o o o X | | o o o o o o o o o o X |
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| o o o o o o o o o o Y | | o o o o o o o o o o Y |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| + | </pre> |
| | | |
| The local flag E_2@Y is displayed here: | | The local flag E_2@Y is displayed here: |
| | | |
| + | <pre> |
| 0 1 2 3 4 5 6 7 8 9 | | 0 1 2 3 4 5 6 7 8 9 |
| o o o o o o o o o o X | | o o o o o o o o o o X |